<!DOCTYPE html>
<html lang="en-US">
<!--********************************************-->
<!--*       Generated from PreTeXt source      *-->
<!--*                                          *-->
<!--*         https://pretextbook.org          *-->
<!--*                                          *-->
<!--********************************************-->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="robots" content="noindex, nofollow">
</head>
<body class="ignore-math">
<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Representing <dfn class="terminology">periodic</dfn> functions as a series of sine and cosine functions.<dfn class="terminology">Definition</dfn> A function <span class="process-math">\(f(x)\)</span> is <dfn class="terminology">periodic</dfn> with period <span class="process-math">\(T\text{,}\)</span> if</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
f(x+T)=f(x),\quad \forall x
\end{equation*}
</div>
<p class="continuation">Note: If <span class="process-math">\(T\)</span> is a period, so are <span class="process-math">\(2T, 3T, \cdots\text{.}\)</span>The smallest period <span class="process-math">\(T\)</span> is called the <dfn class="terminology">fundamental period</dfn>.<dfn class="terminology">Properties:</dfn> If <span class="process-math">\(f(x)\)</span> and <span class="process-math">\(g(x)\)</span> are both periodic with period <span class="process-math">\(T\text{,}\)</span> so will any linear combination <span class="process-math">\(af(x) + bg(x)\)</span> for <span class="process-math">\(a, b\in\mathbb{R}\text{;}\)</span> Also, the product <span class="process-math">\(f(x)g(x)\)</span> is periodic with the same period <span class="process-math">\(T\text{.}\)</span>Known examples of periodic functions: trig functions.With period <span class="process-math">\(2\pi\text{:}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\sin x, \sin 2x, \sin 3x, \cdots ,\cos x, \cos2x, \cos3x,\cdots
\end{equation*}
</div>
<p class="continuation">With period <span class="process-math">\({2L}\text{:}\)</span>  (<span class="process-math">\(L&gt;0\)</span>)</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\sin \frac{\pi x}{L}, \sin \frac{2\pi x}{L}, \sin \frac{3\pi x}{L}, \cdots, \cos \frac{\pi x}{L}, \cos \frac{2\pi x}{L}, \cos \frac{3\pi x}{L}, \cdots
\end{equation*}
</div>
<p class="continuation">We have the <dfn class="terminology">trig set</dfn>:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\left\{1, \sin \frac{m\pi x}{L}, \cos \frac{m\pi x}{L}\right\},\qquad m=1,2,\cdots
\end{equation*}
</div>
<p class="continuation"><dfn class="terminology">Definition</dfn> Let <span class="process-math">\(f(x)\)</span> be periodic with period <span class="process-math">\(2L\text{.}\)</span> <dfn class="terminology">Fourier series</dfn> for <span class="process-math">\(f(x)\)</span> is:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation}
\hat{f}(x)=\frac{a_0}{2}+\sum_{m=1}^{\infty}\left(a_m\cos\frac{m\pi x}{L} +b_m\sin\frac{m\pi x}{L}\right)\tag{7.3.1}
\end{equation}
</div>
<p class="continuation">Here the constants <span class="process-math">\(a_0\text{,}\)</span> <span class="process-math">\(a_m\text{,}\)</span> <span class="process-math">\(b_m\)</span> are called: <dfn class="terminology">Fourier coefficients</dfn>.</p>
<span class="incontext"><a href="sec7_3.html#p-339" class="internal">in-context</a></span>
</body>
</html>
